What is the answer to (z-1)/z =3+4i ?

The answer to (z-1)/z =3+4i is z=-1/10+1/5 i

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(z-1)/z =3+4i

\frac{z - 1}{z} = 3 + 4 i

z = - \frac{1}{10} + \frac{1}{5} i

Solution steps

\frac{z - 1}{z} = 3 + 4 i

Substitute

z = x + y i

\frac{x + y i - 1}{x + y i} = 3 + 4 i

Multiply both sides by

x + y i

\frac{x + y i - 1}{x + y i} (x + y i) = 3 (x + y i) + 4 i (x + y i)

Expand

(x - 1) + i y = (3 x - 4 y) + i (3 y + 4 x)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[x - 1 = 3 x - 4 y y = 3 y + 4 x]

[x - 1 = 3 x - 4 y y = 3 y + 4 x] : x = - \frac{1}{10}, y = \frac{1}{5}

Substitute back

z = x + y i

z = - \frac{1}{10} + \frac{1}{5} i

25 - m^{2} = (- 1)^{2} + (mi + 2 i)^{2}

\frac{z - 5 i}{z + 2} = 4 i

3 x + 2 y i - i x + 5 y = 7 + 5 i

(x - 2 y i) - (y - x i) = 2 + i

z - (\frac{1}{z}) = 1 + 5 i

What is the answer to (z-1)/z =3+4i ?
The answer to (z-1)/z =3+4i is z=-1/10+1/5 i